JournalsjcaVol. 4, No. 4pp. 369–395

An explicit construction of the universal division ring of fractions of Ex1,,xdE\langle\langle x_1,\ldots, x_d\rangle \rangle

  • Andrei Jaikin-Zapirain

    Universidad Autónoma de Madrid, Spain
An explicit construction of the universal division ring of fractions of $E\langle\langle x_1,\ldots, x_d\rangle \rangle$ cover
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Abstract

We give a sufficient and necessary condition for a regular Sylvester matrix rank function on a ring RR to be equal to its inner rank ρR\rho_R. We apply it in two different contexts.

In our first application, we reprove a recent result of T. Mai, R. Speicher and S. Yin: if X1,,XdX_1,\ldots, X_d are operators in a finite von Neumann algebra M\mathcal M with a faithful normal trace τ\tau, then they generate the free division ring on X1,,XdX_1,\ldots, X_d in the algebra of unbounded operators affiliated to M\mathcal M if and only if the space of tuples (T1,,Td)(T_1,\ldots, T_d) of finite rank operators on L2(M,τ)L^2(\mathcal M,\tau) satisfying

i=1d[Tk,Xk]=0,\sum_{i=1}^d [T_k,X_k]=0,

is trivial.

In our second and main application we construct explicitly the universal division ring of fractions of Ex1,,xnE\langle\langle x_1,\ldots, x_n\rangle\rangle, where EE is a division ring, and we use it in order to show the following instance of pro-pp Lück approximation.

Let FF be a finitely generated free pro pp-group, F=F1>F2>F=F_1 > F_2 > \cdots a chain of normal open subgroups of FF with trivial intersection and AA a matrix over Fp[[F]]\mathbb F_p [[F]]. Denote by AiA_i the matrix over Fp[F/Fi]\mathbb F_p[F/F_i] obtained from the matrix AA by applying the natural homomorphism Fp[[F]]Fp[F/Fi]\mathbb F_p [[F]] \to \mathbb F_p[F/F_i]. Then there exists the limit

limirkFp(Ai)F:Fi\displaystyle \lim_{i\to \infty} \frac{\mathrm {rk}_{\mathbb F_p} (A_i)}{|F:F_i|}

and it is equal to the inner rank ρFp[[F]](A)\rho_{\mathbb F_p [[F]]}(A) of the matrix AA.

Cite this article

Andrei Jaikin-Zapirain, An explicit construction of the universal division ring of fractions of Ex1,,xdE\langle\langle x_1,\ldots, x_d\rangle \rangle. J. Comb. Algebra 4 (2020), no. 4, pp. 369–395

DOI 10.4171/JCA/47